Further results on snakes in powers of complete graphs

AbstractBy a snake in a finite graph G is meant a cycle without chords. Denoted by S(G) the length of a longest snake in G. In this paper we obtain a new lower bound for S(G) in the case where G is the product of d copies of the complete graph on n vertices

On the snake in the box problem

AbstractA snake in a graph is a simple cycle without chords. Denote by s(d) the length of a longest snake in the d-dimensional unit cube. We give a new proof of the theorem of Evdokimov that s(d) > λ2d, where λ is a positive constant

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

From Nonspecific DNA–Protein Encounter Complexes to the Prediction of DNA–Protein Interactions

©2009 Gao, Skolnick. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.doi:10.1371/journal.pcbi.1000341DNA–protein interactions are involved in many essential biological activities. Because there is no simple mapping code between DNA base pairs and protein amino acids, the prediction of DNA–protein interactions is a challenging problem. Here, we present a novel computational approach for predicting DNA-binding protein residues and DNA–protein interaction modes without knowing its specific DNA target sequence. Given the structure of a DNA-binding protein, the method first generates an ensemble of complex structures obtained by rigid-body docking with a nonspecific canonical B-DNA. Representative models are subsequently selected through clustering and ranking by their DNA–protein interfacial energy. Analysis of these encounter complex models suggests that the recognition sites for specific DNA binding are usually favorable interaction sites for the nonspecific DNA probe and that nonspecific DNA–protein interaction modes exhibit some similarity to specific DNA–protein binding modes. Although the method requires as input the knowledge that the protein binds DNA, in benchmark tests, it achieves better performance in identifying DNA-binding sites than three previously established methods, which are based on sophisticated machine-learning techniques. We further apply our method to protein structures predicted through modeling and demonstrate that our method performs satisfactorily on protein models whose root-mean-square Ca deviation from native is up to 5 Å from their native structures. This study provides valuable structural insights into how a specific DNA-binding protein interacts with a nonspecific DNA sequence. The similarity between the specific DNA–protein interaction mode and nonspecific interaction modes may reflect an important sampling step in search of its specific DNA targets by a DNA-binding protein

THUMP from archaeal tRNA:m(2)(2)G10 methyltransferase, a genuine autonomously folding domain

The tRNA:m(2)(2)G10 methyltransferase of Pyrococus abyssi (PAB1283, a member of COG1041) catalyzes the N(2),N(2)-dimethylation of guanosine at position 10 in tRNA. Boundaries of its THUMP (THioUridine synthases, RNA Methyltransferases and Pseudo-uridine synthases)—containing N-terminal domain [1–152] and C-terminal catalytic domain [157–329] were assessed by trypsin limited proteolysis. An inter-domain flexible region of at least six residues was revealed. The N-terminal domain was then produced as a standalone protein (THUMPα) and further characterized. This autonomously folded unit exhibits very low affinity for tRNA. Using protein fold-recognition (FR) methods, we identified the similarity between THUMPα and a putative RNA-recognition module observed in the crystal structure of another THUMP-containing protein (ThiI thiolase of Bacillus anthracis). A comparative model of THUMPα structure was generated, which fulfills experimentally defined restraints, i.e. chemical modification of surface exposed residues assessed by mass spectrometry, and identification of an intramolecular disulfide bridge. A model of the whole PAB1283 enzyme docked onto its tRNA(Asp) substrate suggests that the THUMP module specifically takes support on the co-axially stacked helices of T-arm and acceptor stem of tRNA and, together with the catalytic domain, screw-clamp structured tRNA. We propose that this mode of interactions may be common to other THUMP-containing enzymes that specifically modify nucleotides in the 3D-core of tRNA

Crystal structures of the CusA efflux pump suggest methionine-mediated metal transport

Gram-negative bacteria, such as Escherichia coli, frequently utilize tripartite efflux complexes in the resistance-nodulation-cell division (RND) family to expel diverse toxic compounds from the cell.1,2 The efflux system CusCBA is responsible for extruding biocidal Cu(I) and Ag(I) ions.3,4 No prior structural information was available for the heavy-metal efflux (HME) subfamily of the RND efflux pumps. Here we describe the crystal structures of the inner membrane transporter CusA in the absence and presence of bound Cu(I) or Ag(I). These CusA structures provide important new structural information about the HME sub-family of RND efflux pumps. The structures suggest that the metal binding sites, formed by a three-methionine cluster, are located within the cleft region of the periplasmic domain. Intriguingly, this cleft is closed in the apo-CusA form but open in the CusA-Cu(I) and CusA-Ag(I) structures, which directly suggests a plausible pathway for ion export. Binding of Cu(I) and Ag(I) triggers significant conformational changes in both the periplasmic and transmembrane domains. The crystal structure indicates that CusA has, in addition to the three-methionine metal binding site, four methionine pairs - three located in the transmembrane region and one in the periplasmic domain. Genetic analysis and transport assays suggest that CusA is capable of actively picking up metal ions from the cytosol, utilizing these methionine pairs/clusters to bind and export metal ions. These structures suggest a stepwise shuttle mechanism for transport between these sites

FitEM2EM—Tools for Low Resolution Study of Macromolecular Assembly and Dynamics

Studies of the structure and dynamics of macromolecular assemblies often involve comparison of low resolution models obtained using different techniques such as electron microscopy or atomic force microscopy. We present new computational tools for comparing (matching) and docking of low resolution structures, based on shape complementarity. The matched or docked objects are represented by three dimensional grids where the value of each grid point depends on its position with regard to the interior, surface or exterior of the object. The grids are correlated using fast Fourier transformations producing either matches of related objects or docking models depending on the details of the grid representations. The procedures incorporate thickening and smoothing of the surfaces of the objects which effectively compensates for differences in the resolution of the matched/docked objects, circumventing the need for resolution modification. The presented matching tool FitEM2EMin successfully fitted electron microscopy structures obtained at different resolutions, different conformers of the same structure and partial structures, ranking correct matches at the top in every case. The differences between the grid representations of the matched objects can be used to study conformation differences or to characterize the size and shape of substructures. The presented low-to-low docking tool FitEM2EMout ranked the expected models at the top